let-z-from-C-prove-that-arcsinz-iln-iz-1-z-2-arccosz-iln-z-z-2-1-

Question Number 73473 by mathmax by abdo last updated on 13/Nov/19

l e t z f r o m C p r o v e t h a t

a r c s i n z = - i l n (i z + \sqrt{1 - z^{2}})

a r c c o s z = - i l n (z + \sqrt{z^{2} - 1})

Commented by mathmax by abdo last updated on 15/Nov/19

w e h a v e \frac{d}{d z} (a r c s i n z) = \frac{1}{\sqrt{1 - z^{2}}} a n d \frac{d}{d z} (- i l n (i z + \sqrt{1 - z^{2}})

= - i \times \frac{i - \frac{2 z}{2 \sqrt{1 - z^{2}}}}{i z + \sqrt{1 - z^{2}}} = - i \times \frac{\frac{i \sqrt{1 - z^{2}} - z}{\sqrt{1 - z^{2}}}}{i z + \sqrt{1 - z^{2}}} = \frac{i z + \sqrt{1 - z^{2}}}{\sqrt{1 - z^{2}} (i z + \sqrt{1 - z^{2}})}

= \frac{1}{\sqrt{1 - z^{2}}} \Rightarrow a r c s i n z = - i l n (i z + \sqrt{1 - z^{2}}) + c

z = 0 \Rightarrow 0 = - i l n (1) + c = 0 + c \Rightarrow c = 0 \Rightarrow a r c s i n (z) = - i l n (i z + \sqrt{1 - z^{2}})

a n d w e u s e t h e s a m e m e t h o d f o r

a r c o s z = - i l n (z + \sqrt{z^{2} - 1}) .